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8. Review of PDEsPlease make your own overview of PDE including the topics andmethods we have discussed in class.Remark: You may use LaTeX or graphic software to vi...

Question

8. Review of PDEsPlease make your own overview of PDE including the topics andmethods we have discussed in class.Remark: You may use LaTeX or graphic software to visualize.Include graphs of common PDEs and characteristics.Thank you for the help I will rate

8. Review of PDEs Please make your own overview of PDE including the topics and methods we have discussed in class. Remark: You may use LaTeX or graphic software to visualize. Include graphs of common PDEs and characteristics. Thank you for the help I will rate



Answers

In Problems 1 - 12, a differential equation is given along with the field or problem area in which it arises. Classify each as an ordinary differential equation (ODE) or a partial differential equation (PDE), give the order, and indicate the independent and dependent variables. If the equation is an ordinary differential equation, indicate whether the equation is linear
or nonlinear.
$$\begin{array}{l}{\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0} \\ {\text { (Laplace's equation, potential theory, electricity, heat, }} \\ {\text { aerodynamics) }}\end{array}$$

This is problem five from one point run and given this equation fresh aggression, White times one applies theory to do I d. X, which is doing it off. Why, with respect X Square? Yes, it is equal to or some constant C. Now. First question is whether this is no tea or pity. And since we see that there's just received, there's delight the extreme here, which means that wise thing, the bend dependent variable and access the deep end invariable on because we only see there's because one is the only one dependent variable. This said this would be nobody. So what this means? It said, Why is dependent? I'm gonna a previous year, a little bit dependent variable and thanks is independent variable far in creation. And since there's only one independent variable, I'm going the equation I'm going, which I'm gonna call, which I will All this aggression c creation one. And so I could say one is okay now, next, we want to know if this odious, linear or long, nonlinear and again our equation that is implicit in the sense that I could write. This s why Waas one minus my prime square trying to see staying Quarter zero and we determine the linear plane. Yeah, return on the narrative this equation By looking at this function here which I will call half off X Y and white pine. Now, as a function of these variables, the question we ask ourselves this is this Is this a linear? And obviously we have this y prime square term. Therefore, our equation, it's non linear. And finally the order of the question would be just one. Because by definition, the order is tthe e highest order of to do. Wouldn't do that it Pierre skating. In our case, we have Onley wine Prime appears so This would be our s o Our equation would be also first order.

And this problem, we want to classify the given differential equation route one minus Y squared. Why the export was to F. T. Y. Yet? Because, you know, this question is showing your basic understanding of differential equation in order to solve it is imperative that we have an understanding of what type of differential equation we're solving. That's always critical to using certain techniques to solve differential equations. That this question is just simply evaluating whether or not we can identify the proper wages were being asked to solve. So that's the kind of question with the answer is what kind of equation of this ordinary? A partial? What is the order of the differential equation? Is it linear or nonlinear? Ordinary and so on. To start with you? The only european. So it's ordinary are part of the equation because all differential don't expect the X. There are no parts of the rivers. So is it linear? Nonlinear? It's not linear because one wonders why the nominee turn. So what is the order that you think you We use? The order of the highest order derivative that's described by the X ray? We see this differential equation, the second word. Finally, we classify the variables since the function why it's being differentiated with respect to X. We assume why the function of X. So X. Is the independent variable why the dependent variable.

Hello, everyone. Today we will be classifying the equation eight times. D for y d x to the fourth equals X times one minus X. I noticed that there is a single ordinary derivative, and so we get right away that this is an ordinary differential equation. Now, at the same time, we easily get the order of the equation because we have 1/4 derivative here. And so we can conclude that this is 1/4 order differential equation. Now to determine the independent variables in the dependent variables, recall that for a derivative D. Y t X determined, the top determines the dependent variable at the term at the bottom determines the independent variable. And so, in the case of our equation, we have that X is the independent variable. And why is the dependent variable finally, to determine whether the equation ist linear or non been year. We need to look at all the terms that have y in it and all of its derivatives. We consider any Constance as functions of the independent variable, and so in this case we have eight, which is a function of X, and we have X times one minus x And so it's assistance of function on Lee of the Independent Variable an aide is a function that can be treated as a function of only the independent variable. This satisfies the conditions for this to be 1/4 order linear, ordinary differential equation.

Which is problem nine from one by one in which we have a differential equation will accept this. X Do you are day squared? Y The X word Plus, Do you like the X plus X times? Well, it's difficult to see her No person. I'm not a big fan of this, and I still like the exact vision so we can write this a question. Similarly, just by using the planet shacks times right out with crime and then plus why Single prime was X times. Why this vehicle to zero now it just looks a little better. And we want to know if it's no TV or beauty and, well, there's only one day with one kind of there to hear. Why do it off? Why, with respect X and X. So it's only one independent variable. So this means let's call our equation again as usual, Question of one. So one is iss o d. And just think we've just like you mentioned X is in the variable while why it's the D pendant for able, because why depends on X and the excellent Next? We want to know if the equation is linear or nothing here And for that we just look at this, this here. And if he again to formally we write this as a function off all independent variable and dependent variable and all efforts to it. So this is F of X. Why my prime and white double crime now for the moment, forget about the there with theirs and functions and just look at the form of this equation. And we see that this is a linear equation in terms off the Y variables. Because, um, Thekla visions off each white terms or just X functions of ex is here. There's one condition here. There's one cushion here, and there's simply there's just here. It's just one times wine prime, and the key idea is the wide terms do not interact with each other in a non linear way. In other words, stood on like want to play together. Our Duce knows there's no square off them, so this means that this is a linear creation, this dysfunction f It's a linear function after my variables. This means that our equation is what's called a linear of the so that appears it's Langer. So basically the key observation have to make is whether there is any interaction between the white terms other than adding and subtracting. And so this is linger. And what about the order? Well, we have. Why? The highest order we have in terms of through there is here. All right. Why too wide? Double prime. So this means that our immigration is off, Singer.


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